The Beginnings and Evolution of Algebra

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How did algebra arise? What are its subject matter and methods? How have they changed in the process of its evolution? Authors I. G. Bashmakova and G. S. Smirnova answer these questions in an authoritative and compelling way in this latest addition to the Dolciani Mathematical Expositions Series.

There is hardly a branch of mathematics whose evolution has undergone as many surprising metamorphoses as has algebra, and these metamorphoses are described by the authors with vividness and clarity. The special merit of the book is that it corrects the widespread view that up to the 1830s the mainspring of the development of algebra was the investigation and solution of determinate algebraic equations, and especially their solution by radicals. The authors show that this viewpoint is one-sided and gives a distorted view of the evolution of algebra. Specifically, they show that the role of indeterminate equations in the evolution of algebra was no less important than that of determinate equations.

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Editorial Reviews

Charles Ashbacher
"As is so well documented in this book, the concrete applications of geometry was the first of the mathematical arts, closely followed by algebra. While necessary to measure the physical world, geometry as we now know it would have been very limited without the ability to express the ideas in symbols. The authors also convincingly argue that the solving of indeterminate equations was no less significant than the solving of determinate equations in algebra...I found this book captivating as the authors present in great detail how algebra evolved from the first primitive steps to the dynamic and encompassing entity that it is today. Every mathematician should take some time to read this book and appreciate what their predecessors did."
Charles Ashbacher Technologies
"The book effectively conveys the piecemeal, often idiosyncratic, way in which algebra evolved, including the changes of emphasis and notation that have aided or impeded progress.... Highly recommended. Upper-division undergraduates through faculty."
G. L. Alexanderson
"In addition to having one of the most beautiful book covers of this or any year, this book presents the history of algebra with great clarity and elegance-with no small amount of credit to Shenitzer, who provided a smooth and idiomatic translation from the Russian.... The book shows us a splendid panorama of the development of a discipline with a fascinating history stretching over four millennia. It's a winner!"
in MAA Online
Jeanne Ramirez
"It is fascinating to read about the historical context of mathematical developments, the tidbits of personal history about mathematical developments, the tidbits of personal history about mathematicians, the development of algebraic notation, and the mathematical insights that are often overlooked for centuries. The authors investigate the relationship between algebra and geometry throughout history; they give extensive coverage to Diophantus, Gauss, Descartes, Galois, and Euler."
The Mathematics Teacher
Steve Abbott
"The authors clearly aimed all along to appeal to mathematicians. The treatment is genuinely expository and accessible to undergraduates who are studying the corresponding mathematics.... Books like this serve a useful purpose in making it clear that mathematics starts from concrete problems whose generalization reveals the underlying theory only gradually. I gladly recommend The Beginnings and Evolution of Algebra as a companion to undergraduate courses."
Mathematical Gazette
A history of algebra, starting with an account of its Babylonian beginnings and ending with a substantial account of major developments in the 19th century. While early chapters can be read by people with no more than a high school knowledge of mathematics, later chapters require familiarity with material usually presented in a course in abstract algebra. Bashmakova teaches in the department of mechanics and mathematics at Moscow State University. Smirnova has taught at Moscow State University since 1987. Member price $19.95. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780883853290
  • Publisher: Mathematical Association of America
  • Publication date: 1/1/2000
  • Series: Dolciani Mathematical Expositions Series
  • Edition description: New Edition
  • Pages: 196
  • Product dimensions: 5.90 (w) x 8.90 (h) x 0.50 (d)

Read an Excerpt

The 19th Century witnessed fundamental transformations of the major divisions of mathematics-geometry, algebra, and mathematical analysis. Of these, the qualitative changes in geometry, especially the creation of non-Euclidean and multidimensional geometries, may well have had the profoundest effect on the mathematical imagination. As a result-as noted by Bourbaki- classical geometry became a universal language for the interpretation of mathematical facts and theories. One could talk of a geometric style of thinking (A.N. Kolmogorov, The Great Soviet Encyclopedia, 2nd ed., Vol. 26 (Russian)).
 But at the end of the 20th century it became clear that one would be equally justified in calling the 19th century the age of new algebra, and in talking of the algebraization of mathematics and the elaboration of an algebraic style of thinking. Indeed, until the 19th century algebra was largely the science of (determinate and indeterminate) equation, whereas in the 19th century there appeared in it completely new concepts and objects, such as resulted in the development of new methods and conceptions, and this brought about a changed view of the subject matter of algebra. Specifically, the task which there are defined operations with properties more or less similar to those of addition and multiplication of numbers" (A.D. Kurosh and O. Yu. Schmidt, The Great Soviet Encyclopedia, 2nd ed., Vol. 1 (Russian)). These operations were called composition laws and their basic properties were given by systems of axioms.
 The methods of this new, so called "modern", algebra quickly entered other areas of mathematics. The 19th century witnessed the construction of algebraic number theory, the development of the first stages of algebraic geometry, and the rigorization of the theory of Riemann surfaces by algebraic means. But what was startling was the "victorious march" of group theory, which is today an indispensable ingredient of every area of mathematics. We will talk about this in detail in Chapter VII (5).
 We add that functional analysis, which came into being at the end of the 19th century, was constructed as linear algebra of infinite-dimensional spaces.
 Finally, from the first quarter of the 20th century onward, algebraic methods were intensively applied in physics and brought about its radical transformation. There came into being matrix mechanics, the theory of spinors, and the subject of symmetry, which plays so important a role today. Already in the 1890's E.S. Fedorov applied the theory of finite groups to crystallography, and in this way managed to solve the problem of classification of regular point systems in space (we recall that there are 230 Fedorov space groups and that their classification could not be carried out without the use of group theory). Similarly, more general groups and their representations are used to classify elementary particles and their motions.
 How did algebra arise? What are its subject matter and methods? How have they changed in the process of its evolution? These are the questions we will try to answer in the present essay. Before we do so we note that the view of algebra as the science of operations defined on sets of arbitrary objects came into being quite late, probably only in the 1930's.
 In its evolution algebra passed through different phases during which it was thought of differently. Views of its subject matter, methods, and aims changed. There is hardly a branch of mathematics whose evolution involved as many surprising metamorphoses as that of algebra. Nevertheless, is we cast a retrospective glance at its development, then we see that the characteristic feature of algebra from its very first steps and practically until the appearance (at the beginning of the 19th century) of noncommutative and nonassociative systems was the study of laws of composition and of their fundamental properties: commutativity of addition and multiplication, distributivity of multiplication over addition, rules for multiplication of binomials and for raising them to powers, rules for operating with equations, and so on. This being so, we will begin our study of the history of algebra from the time of the discovery and application of the simplest laws of composition.
 When characterizing each of the fundamental stages of the evolution of algebra we will focus our attention on the problems that faced it and stimulated its development as well as on the basic ideas and methods used in their investigation.
 Modern history of mathematics seems to be dominated by the view that up to the 1830s the mainspring of the development of algebra was the investigation and solution of determinate algebraic equations, and especially their solution by radicals. We will show that this viewpoint is one-sided and gives a distorted representation of its evolution. In short, we claim that the role of indeterminate equations in the development of algebra was no less important than that of determinate equations.
 We note also that the rate and phases of the evolution of algebra do not always correspond to the rate and periods of evolution of mathematics as a whole. In our account the history of algebra is divided into the following basic stages.
1. Numerical algebra of ancient Babylonia (a phase that coincides in time with the first period of history of mathematics i.e., the period of accumulation of mathematical knowledge).
2. Geometric algebra of classical antiquity (a phase lasting from the 5th to the 1st century BCE, which corresponds to the first half of the second period of the history of mathematics, the period of the transformation of mathematics into an abstract theoretical science).
3. The rise of literal algebra (from its birth to the creation of literal calculus; this phase began in the 1st century CE and lasted until the end of the 16th century, i.e., it began in the second half of the second period of the history of mathematics and lasted until the end of its third period, the period of development of elementary mathematics).
4. Creation of the theory of algebraic equations (a phase that comprises the development of algebra in the 17th and 18th centuries and ends in the 1830s).
5. Formation of the foundations of modern algebra (a phase lasting from the 1830s and the 1930s).
 We will provide detailed characterizations of these phases in parallel with the exposition of the relevant material. The last phase of the evolution of algebra, the one that began some fifty odd years ago, cannot as yet be classified as a component of history.
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Table of Contents


Chapter 1: Elements of Algebra in ancient Babylonia
1. Babylonian numerical algebra
2. Indeterminate equations
3. The origin of the first algebraic problems

Chapter 2: Ancient Greek "geometric algebra"
1. Transformation of mathematics into an abstract deductive science. Discovery of incommensurability
2. Geometric algebra
3. Classification of quadratic irrationalities
4. The first unsolvable problems
5. Cubic equations
6. Indeterminate equations

Chapter 3: The birth of literal algebra
1. Mathematics in the first centuries CE Diophantus
2. Diophantus Arithmetica, Its domain of numbers and symbolism
3. The contents of Arithmetica. Diophantus' methods
4. Algebra over Diphantus

Chapter 4: Algebra in the Middle Ages in the Arabic East and in Europe
1. The emergence of algebra as an independent discipline
2. The first advances in algebra in Europe
3. Algebraic symbolism in Europe. The German cossists and the development of algebra in Italy
Chapter 5: The first achievements of algebra in Europe
1. The solution of cubic and quartic equations
2. The Algebra of Rafael Bombelli. Introduction of complex numbers
3. Francois Viete
4. Creation of a literal calculus
5. Genesis triangulorum
6. Indeterminate equations in the work of Viete
7. Beginning of the theory of determinate equations

Chapter 6: Algebra in the 17th and 18th centuries
1. The arithmetization of algebra
2. Descartes' treatment of determinate equations
3. The fundamental theorem of algebra
4. Gauss' criticism
5. The problem of solution of equations by radicals
6. Proof of the unsolvability of the general quintic by radicals

Chapter 7: The theory of algebraic equations in the 19th century
1. Cyclotomic equations
2. Equations with an Abelian group
3. Galois theory
4. The evolution of group theory in the 19th century
5. The victorious "march" of group theory

Chapter 8: Problems of number theory and the birth of commutative algebra
1. Diophantine equations and the introduction of algebraic numbers
2. Kummer's ideal factors
3. Arithmetic in arbitrary fields of algebraic numbers, Ideal theory
4. Ideal theory in fields of algebraic functions

Chapter 9: Linear and noncommutative algebra
1. Introduction of determinants
2. Linear transformations and matrices
3. The English school of symbolic algebra, Hamilton's quaternions
4. Algebras

Appendix: A new view of the geometric algebra of the ancients


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